520 ON THE SCATTERING OF LIGHT BY SPHEEICAL SHELLS [427
BB parallel to A A at a distance £ the linear retardation is — 2^ cos |^, as in the theory of thin plates; and the exponential factor is e-^pe-ik^cos^x. The
elementary volume at BB is still expressed by 'ZtrRdRd^, and accordingly
The integral in (8) is 2R sin m/m, m being given by (6), and we recover (7) as expressing the value of dP for a spherical shell of volume
The value of dP for a spherical shell having been now obtained independently, we can pass at once by integration to the corresponding expression for a complete sphere of uniform optical quality, thus recovering (5) by a simpler method not involving Bessel's functions at all. And a comparison of the two processes affords a demonstration of Hobson's theorem formerly employed as a stepping stone.
When P is known, the secondary vibration is given by (2), in which we may replace r by p. So far as it depends upon P, the angular distribution, being a function of %, is symmetrical round Ox, the direction of primary propagation. So far as it depends on the other factors ay/p*, etc., it is the same as for an infinitely small sphere ; in particular no ray is emitted in the direction denned by o = /9 = 0, that is in the direction of primary vibration. There is no limitation upon the value of R if (K— 1) be small enough; but the reservation is important, since it is necessary that at every point of the obstacle the retardation of the primary waves due to the obstacle be negligible.
When R is great compared with X, (= 27T/&), m usually varies rapidly with R or Jc, and so does P, as given for the complete uniform sphere in (5). An exception occurs when % is nearly equal to TT, that is when the secondary rays anticipation was easily confirmed.